
Univariate GLM is the general linear model now often used to implement such longestablished statistical procedures as regression and members of the ANOVA family. It is "general" in the sense that one may implement both regression and ANOVA models. One may also have fixed factors, random factors, and covariates as predictors. Also, in GLM one may have multiple dependent variables, as discussed in a separate section on multivariate GLM and one may have linear transformations and/or linear combinations of dependent variables. Moreover, one can apply multivariate tests of significance when modeling correlated dependent variables, not relying on individual univariate tests as in multiple regression. GLM also handles repeated measures designs. Finally, because GLM uses a generalized inverse of the matrix of independent variables' correlations with each other, it can handle redundant independents which would prevent solution in ordinary regression models.
Data requirements. In all GLM models, the dependent(s) is/are continuous. The independents may be categorical factors (including both numeric and string types) or quantitative covariates. Data are assumed to come from a random sample for purposes of significance testing. The variance(s) of the dependent variable(s) is/are assumed to be the same for each cell formed by categories of the factor(s) (this is the homogeneity of variances assumption).
Regression in GLM is simply a matter of entering the independent variables as covariates and, if there are sets of dummy variables (ex., Region, which would be translated into dummy variables in OLS regression, for ex., South = 1 or 0), the set variable (ex., Region) is entered as a fixed factor with no need for the researcher to create dummy variables manually. The b coefficients will be identical whether the regression model is run under ordinary regression (in SPSS, under Analyze, Regression, Linear) or under GLM (in SPSS, under Analyze, General Linear Model, Univariate). Where b coefficients are default output for regression in SPSS, in GLM the researcher must ask for "Parameter estimates" under the Options button. The Rsquare from the Regression procedure will equal the partial Eta squared from the GLM regression model.
The advantages of doing regression via the GLM procedure are that dummy variables are coded automatically, it is easy to add interaction terms, and it computes etasquared (identical to Rsquared when relationships are linear, but greater if nonlinear relationships are present). However, the SPSS regression procedure would still be preferred if the reseacher wishes output of standardized regression (beta) coefficients, wishes to do multicollinearity diagnostics, or wishes to do stepwise regression or to enter independent variables hierarchically, in blocks. PROC GLM in SAS has a greater range of options and outputs (SAS also has PROC ANOVA, but it handles only balanced designs/equal group sizes).
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GLM UNIVARIATE: GENERAL LINEAR MODELS Table of Contents Overview 11 Key Concepts 15 Why testing means is related to variance in analysis of variance 15 Oneway ANOVA 16 Simple oneway ANOVA in SPSS 16 Simple oneway ANOVA in SAS 20 Twoway ANOVA 23 Twoway ANOVA in SPSS 24 Twoway ANOVA in SAS 27 Multivariate or nway ANOVA 29 Regression models 29 Parameter estimates (b coefficients) for factor levels 31 Parameter estimates for dichotomies 32 Significance of parameter estimates 32 Research designs 32 Betweengroups ANOVA design 32 Completely randomized design 34 Full factorial ANOVA 34 Balanced designs 35 Latin square designs 36 GraecoLatin square designs 37 Randomized Complete Block Design (RCBD ANOVA) 37 Split plot designs 39 Mixed design models 39 Random v. fixed effects models 41 In SPSS 41 In SAS 42 Linear mixed models (LMM) vs. general linear models (GLM) 43 Effects 43 Treating a random factor as a fixed factor 44 Mixed effects models 44 Nested designs 44 Nested designs 45 In SPSS 46 In SAS 49 Treatment by replication design 49 Withingroups (repeated measures) ANOVA designs 49 Counterbalancing 50 Reliability procedure 51 Repeated measures GLM in SPSS 51 Repeated measures GLM in SAS 51 Interpreting repeated measures output 52 Variables 53 Types of variables 53 Dependent variable 53 Fixed and random factors 54 Covariates 54 WLS weights 54 Models and types of effects 55 Full factorial models 55 Effects 56 Main effects 56 Interaction effects 56 Residual effects 59 Effect size measures 60 Effect size coefficients based on percent of variance explained 60 Partial etasquared 60 Omegasquared 61 Herzberg's R2 62 Intraclass correlation 62 Effect size coefficients based on standardized mean differences 62 Cohen's d 62 Glass's delta 64 Hedge's g 65 Significance tests 65 Ftest 65 Reading the F value 65 Example 1 66 Example 2 66 Significance in twoway ANOVA 67 Computation of F 67 Ftest assumptions 67 Adjusted means 68 Lack of fit test 68 Power level and noncentrality parameter 69 Hotelling's TSquare 70 Planned multiple comparison ttests 70 Simple ttest difference of means 72 Bonferroniadjusted ttest 72 Sidak test 74 Dunnett's test 74 HSU's multiple comparison with the best (MCB) test 74 Posthoc multiple comparison tests 74 The qstatistic 75 Output formats: pairwise vs. multiple range 76 Tests assuming equal variances 76 Least significant difference (LSD) test 76 The FisherHayter test 77 Tukey's test, a.k.a. Tukey honestly significant difference (HSD) test 78 Tukeyb test, a.k.a. Tukey's wholly significant difference (WSD) test 79 SNK or StudentNewmanKeuls test 80 Duncan test 81 Ryan test (REGWQ) 81 The ShafferRyan test 83 The Scheffé test 83 Hochberg GT2 test 85 Gabriel test 87 WallerDuncan test 87 Tests not assuming equal variances 87 Tamhane's T2 test 87 GamesHowell test 88 Dunnett's T3 test and Dunnett's C test 89 The TukeyKramer test 89 The MillerWiner test 89 More than one multiple comparison/post hoc test 89 Example 89 Contrast tests 91 Overview 91 Types of contrasts 92 Deviation contrasts 92 Simple contrasts 92 Difference contrasts 92 Helmert contrasts 92 Repeated contrasts 92 Polynomial contrasts 93 Custom hypothesis tables 93 Custom hypothesis tables index table 93 Custom hypothesis tables 94 Estimated marginal means 96 Overview 96 EMM Estimates table 98 Other EMM output 101 EMM Pairwise comparisons table 101 EMM Univariate tests table 101 Profile plots 101 GLM Repeated Measures 102 Overview 102 Key Terms and Concepts 103 Withinsubjects factor 103 Repeated measures dependent variables 104 Betweensubjects factors 105 Covariates 105 Models 106 Type of sum of squares 107 Balanced vs. unbalanced models 107 Estimated marginal means 108 Pairwise comparisons 109 Statistics options in SPSS 110 Descriptive statistics 110 Hypothesis SSCP matrices 111 Partial etasquared 111 Withinsubjects SSCP matrix and withinsubjects contrast effects. 112 Multivariate tests. 113 Univariate vs. multivariate models 114 Box's M test 115 Mauchly's test of sphericity 115 Univariate tests of withinsubjects effects 116 Parameter estimates 118 Levene's test 119 Spreadversuslevel plots 120 Residual plots 120 Lack of fit test 122 General estimable function 122 Post hoc tests 122 Overview 122 Profile plots for repeated measures GLM 125 Example 125 Contrast analysis for repeated measures GLM 127 Types of contrasts for repeated measures 128 Simple contrasts example 129 Saving variables in repeated measures GLM 130 Cook's distance 131 Leverage values 131 Assumptions 132 Interval data 132 Homogeneity of variances 132 Homogeneity of variance 133 Appropriate sums of squares 137 Multivariate normality 138 Adequate sample size 139 Equal or similar sample sizes 139 Random sampling 139 Orthogonal error 140 Data independence 140 Recursive models 140 Categorical independent variables 140 The independent variable is or variables are categorical. 140 Continuous dependent variables 140 Nonsignificant outliers 140 Sphericity 141 Assumptions related to ANCOVA: 142 Limited number of covariates 142 Low measurement error of the covariate 142 Covariates are linearly related or in a known relationship to the dependent 142 Homogeneity of covariate regression coefficients 143 No covariate outliers 143 No high multicollinearity of the covariates 144 Additivity 144 Assumptions for repeated measures 144 Frequently Asked Questions 145 How do you interpret an ANOVA table? 146 Isn't ANOVA just for experimental research designs? 148 Should I standardize my data before using ANOVA or ANCOVA? 148 Since orthogonality (uncorrelated independents) is an assumption, and since this is rare in real life topics of interest to social scientists, shouldn't regression models be used instead of ANOVA models? 148 Couldn't I just use several ttests to compare means instead of ANOVA? 148 How does counterbalancing work in repeated measures designs? 149 How is F computed in random effect designs? 150 What designs are available in ANOVA for correlated independents? 150 If the assumption of homogeneity of variances is not met, should regression models be used instead? 151 Is ANOVA a linear procedure like regression? How is linearity related to the "Contrasts" option? 151 What is hierarchical ANOVA or ANCOVA? 151 Is there a limit on the number of independents which can be included in an analysis of variance? 152 Which SPSS procedures compute ANOVA? 152 I have several independent variables, which means there are a very large number of possible interaction effects. Does SPSS have to compute them all? 152 Do you use the same designs (between groups, repeated measures, etc.) with ANCOVA as you do with ANOVA? 152 How is GLM ANCOVA different from traditional ANCOVA? 153 What are paired comparisons (planned or post hoc) in ANCOVA? 153 Can ANCOVA be modeled using regression? 153 How does blocking with ANOVA compare to ANCOVA? 153 What is the SPSS syntax for GLM repeated measures? 154 What is a "doubly repeated measures design"? 155 Bibliography 156 Pagecount: 160